INTRODUCTION TO QUANTUM MECHANICS

Transcription

1 A. La Rosa Letre Notes PSU-Physis PH 45 INTRODUCTION TO QUANTUM MECHANICS PART-I TRANSITION from CLASSICAL to QUANTUM PHYSICS CHAPTER CLASSICAL PHYSICS ELECTROMAGNETISM and RELATIITY REIEW,. ELECTROMAGNETISM..A Mawell s Eqations ME..B Conseqenes of the Mawell Eqations..B.a The wave eqation..b.b Light as eletromagneti radiation..b. Independene of the motion of the sore..b.d The Lorentz s Transformation..C The hypothesis of the ether and the notion of absolte veloity..c.a The Mihelson-Morley Eperiment..C.b Lorentz length-ontration hypothesis..c. Transformation of oordinates based on the Lorentz length-ontration hypothesis. SPECIAL THEORY OF RELATIITY..A Newton s Priniple of Relativity..A.a The Galilean Transformation..A.b Eletromagnetism and the Priniple of Relativity..B Einstein s Priniples of Relativity..B.a The priniples of relativity..b.b Conseqenes of Einstein s priniples of relativity..b.b Relationship between the spae-time oordinates in different inertial referene frames..b.b Relationship between the veloities..c Reqired modifiation of the Mehanis laws to make them ompatible with Einstein s relativity priniples...c.a Relativisti Momentm and Energy of a Partile..C.b Derivation of the relativisti mass

2 ..C. Derivation of the relativisti energy..c.d Eqivalene of mass and energy..c.e Relationships involving p, E and v..d For-omponents vetors and symmetry..d.a Transformation of spae-time oordinates..d.b Transformation of energy-momentm oordinates Referenes Rihard Feynman, The Feynman Letres on Physis, olme I, Chapter 5, 6, 7 R. Eisberg and R. Resnik, Qantm Physis, nd Edition, Wiley, 985. Appendi A

3 Chapter CLASSICAL PHYSICS: ELECTROMAGNETISM and RELATIITY. ELECTROMAGNETISM..A Mawell s Eqations ME Treatise on Eletriity and Magnetism, pblished in 873. E ρ ε0 Gass law.b 0 No magneti monopoles E t B 0 B μ 0 ε0 t EjFaraday s Law Ampere s Law modified by Mawell Here, E E, E, E and B B, B, B are the eletromagneti fields. y z y z jthe vetor is the ondtion rrent density in the medim, The salar ρ is the free harge density, jboth and ρ sbjeted to the eqation of ontinity or the onservation of harge epressed as j t 0...B Conseqenes of the Mawell Eqations..B.a The wave eqation j0for the ase and ρ 0, it an be demonstrated that the omponents of the eletromagneti fields satisfy the wave eqation E y v o E t Y 0 where v o μ o ε o 3

4 That is, the fields E and B travel at onstant speed v o...b.b Light as an eletromagneti radiation When Mawell evalated the speed of the eletromagneti waves, he fond, v o μ o ε o 300,000 Km s. This is the speed of light!!! With eitement Mawell wrote: We an sarely avoid the inferene that light onsists in transverse ndlation of the same medim whih is the ase of eletri and magneti phenomena...b. Independene of the motion of the sore If the sore of the distrbane is moving, the emitted light travels at the same speed in any diretion...b.d Lorentz s Transformation H. A. Lorentz notied a remarkable and rios transformation of oordinates that made the ME invariant: ot t o, y y, z z, t 3 o o Lorentz transformation..c The hypothesis of the ether and the notion of absolte veloity At the end of the 9 th entry, it was still oneived that all waves needed a medim to travel. Aordingly, sientist saw a neessity in figring ot a medim in whih light wold propagate. It is in this ontet that the infamos ether was invented as a medim permeating all the spae, and in whih light wold propagate at speed 300,000 Km s. The ether hypothesis, together with the ME oneption on the independene of the light speed relative to the sore motion, reated a possibility to measre the absolte veloity of an objet 4

5 in partilar the Earth relative to the ether, as sggested in Fig.. below. Fig.. The light beam propagates with speed regardless of the motion of the light sore wold be the veloity of light relative to the ether.. If an observer in the ar measred a speed for the light beam, then it wold mean that the speed of the ar is = -...C.a The Mihelson-Morley Eperiment 887 The objetive was to measre the speed of the Earth throgh the hypothetial ether. C C L Apparats mirrors+light sore moving with speed relative to the hypothetial ether. L A B B D D Degree of interferene between these two beams wold depend on Fig.. Shemati of the eperimental arrangement in the Mihelson-Morley eperiment. Assming that is the speed of light relative to the hypothetial ether, we obtain the following epression: Horizontal-beam travel time: From B to D : t L t, whih gives t L From D to B : t L t, whih gives t L L t horizontal t t L 4 5

6 ertial-beam travel time: From B to C : t3 L t3, whih gives t3 L From C to B : same vale as above. Hene L t vertial 5 Notie in 4 and 5 that t [ horizontal t vertial. That is, thorizontal t vertial 6 This differene in time travel that depends on wold reslt in an interferene between the two beams, an information that old then be sed to allate the speed of the Earth relative to the ether. Mihelson and Morley oriented the apparats so that the line BD was nearly parallel to the Earth s motion in its orbits at ertain day and night. The orbital speed is abot 30 Kms, and the apparats was sensitive enogh to detet the orresponding interferene. Bt, it was never was deteted...c.b Lorentz length-ontration onjetre 89 As a way to eplain Mihelson and Morley s pzzling reslt, Lorentz proposed an ad ho hypothesis that bodies ontrat when they are moving, and that the ontration ors only in the diretion they are moving. That wold make t horizontal smaller in the Mihelson and Morley s eperiment. Let L o is the length of a body at rest in the ether referene. Then when it moves with speed along the horizontal ais, its new horizontal length L wold be smaller than L o by a fator This onjetre aonts for the Mihelson and Morley eperiment sine indeed it makes eqal the times t vertial and t horizontal given in 4 and 5. 6

7 t horizontal L [ Lo L o Lo tvertial Ths preditinng the eperimentally obtained eqality between t vertial and t horizontal. This way, Lorentz was saving the hypothesis of the ether...c. Transformation of oordinates based on the Lorentz length-ontration hypothesis Observer O measres the oordinate of a point P with a meter stik. He lays down the stiker times, so he affirms the distane is meters O O P Fig..3 Measring distanes with stiks of different lengths. Observer O, however, onsiders that observer O is sing a foreshortened rler, so the real distane is meters. Sine the observe O has traveled a distane t away from O, observer O wold say that the position of the same point P, measred in the O-oordinates is t, whih leads to t This is the first eqation of the Lorentz transformation given in epression 3 above. 7

8 Still, Lorentz ontration hypothesis was onsidered too artifiial, as the ether hypothesis ontined to get into other ontraditions.. THE SPECIAL THEORY OF RELATIITY The origins of the speial theory of relativity lie in the development of eletromagnetism, from the time when sientist were trying to harmonize eletromagnetism theory and eperiment with the general priniple of relativity. The development passed throgh Mawell, Lorentz, Poinare and Einstein. It was Einstein, in 905, who made the rial generalization to all physial phenomena, not jst eletromagnetism. In 95 Einstein pblished his General Theory of Relativity, whih etends the speial theory of relativity to the ase of gravitation...a Newton s Priniple of Relativity Historially, the priniple of relativity was stated by Newton althogh it has been an hypothesis se in Mehanis sine the days of Copernis, if not before: The motion of bodies inlded in a given spae is the same among themselves, whether that spae is at rest 7 or moves niformly forward in a straight line. In all eperiments performed inside a moving system the laws of physis will appear the same as they wold if the system were standing still. 3 Under what onditions is this priniple valid? Let s review some history...a.a The Galilean Transformation Let s assme that the spatial and temporal oordinates of two referene systems here denominated prime and no prime are related by here o is a onstant = + o t, y=y, z=z, t=t. 8 Galilean Transformation 8

9 Notie that, sine d d y d z d d y d z,,,,, dt dt dt dt dt dt d F d mo, y,z implies F m, y,z dt dt 9 That is, Newton s laws are of the same form in a moving system as in a stationary system. The laws of mehanis Newton s laws then appear in agreement with the priniple of relativity...a.b Eletromagnetism and the Priniple of Relativity The Mawell Eqations ME, however, did not appear to satisfy the priniple of relativity. That is, when sing the Galilean transformation 8, the ME do not remain the same. 4 Conseqently, the predition appeared to be that the eletrial phenomena in a moving system of referene shold be different from those in a stationary referene whih ased the ME into qestion. Eperimental evidene amonted to the ME to prevail. Something was wrong then, and the ME appeared not to be the lprit...b The Speial Theory of Relativity..B.a Einstein s Priniples of Relativity By the end of the 9 th entry, there eisted a few possibilities: 4. The ME were wrong The proper theory of eletromagnetism wold be one that is invariant nder the Galilean transformation.. Galilean transformation applied to lassial mehanis, bt eletromagnetism had a preferred referene frame, the one in whih the ether was at rest. 3. There wold eist a relativity priniple valid for both lassial mehanis and eletromagnetism; bt it was not one in whih the Galilean transformation were valid. This wold imply that the laws of mehanis were in need of modifiation. 9

10 Einstein hose the third option. Einstein s Priniples of Relativity is base on two postlates: The first postlate was enniated already in 7 above. The seond postlate states: The speed of light is finite and independent of the 0 motion of its sore...b.b Conseqenes of Einstein s Priniples of Relativity..B.b Relationship between the spae and times oordinates in different inertial referene frames Consider two inertial referenes that move relative to eah other with onstant veloity. The seond postlate fores a peliar relationship between the spae and times oordinates, y, z, t and, y, z, t sine both referenes have to measre the same speed of light. If at t=0 the two origins O and O oinide and a flash of light is emitted, observer O will note that the light has reahed point A in a time t, and will write r t, or, eqivalently y z t a while observer O will note that the light has reahed the same point A in a time t, and will write r t, or, eqivalently. y z t b 0

11 A Y O O X Y Y O X O r r r =, y, z, t X r =, y, z, t Z Z Z At t=0 At t > 0 o Fig..4 Left: Two referenes synhronize their loks at t=0. Right: Monitoring a light beam from the two different referenes. Assming that the relationship between the oordinates is linear, for eample k t and t a t b, where k, a, and b to be determined from the onditions a and b, the following reslt is obtained, t t, y y, z z, t The transformation given in is nothing bt the Lorentz transformation, mentioned above in 3, nder whih the Mawell Eqations are invariant. In other words, The Mawell s Eqations of eletromagnetism 3 already inherently ontain the relativisti effets. This means that what needs to be hanged are rather the laws of mehanis!! Finally, notie by symmetry that,

12 t t, y y, z z, t 4..B.b Relationship between the veloities The following relationships aqire importane for their role in evalating later in this hapter the linear momentm of a partile in different inertial referenes. The appliation of the onservation of the relativisti linear momentm leads to important and interesting onseqenes as we will see in the net setions. From 4 t t o, y y, z z, t Sine t, t, et, we obtain y z y z eloities transformation 5..C How to modify the laws of lassial mehanis in order to make them ompatible with the Einstein s priniple of relativity In other words, what modifiation shold be introded in order to make the laws of mehanis satisfy the Lorentz transformation? It trns ot as we will demonstrate below that the only reqired ondition is that the mass m assmed to be onstant in the Newton s formlation rather varies with the veloity of the partile.

13 m mo 6 v Einstein s modifiation of Newton s Law That is, the mass m inreases with veloity. m o represents the rest mass the mass of the partile as measred standing still with respet to the observer...c.a Relativisti Momentm and Energy of a Partile The derivation of epression 6 reslts from the reqirement of the following onditions: a ompatibility with the onservation laws of energy and momentm, and b ompatibility with the Lorentz transformation. A generalization of the momentm and energy, onsistent with the Lorentz transformation, beomes neessary. To that effet, the following general epressions are onsidered P Mv v 7 E E v where M and E are salar fntions of the partile s veloity. An elegant and simple derivation for M v is obtained by Feynman whih we reprode in the net setion below; a simple derivation of E v is fond in the book by Eisberg and Resnik. 5 A more formal proedre to obtain M v and E v is presented in Jakson s tetbook. 6 The reslt is, M v m o - v and E E v [ M v. Written in a simpler form: m P o v m v to be jstified below v 8 3

14 m o E v m The energy E m is the total energy of a body...c.b Derivation of the relativisti mass p [mv v Let or, eqivalently, p = [mv v and p y =[mv v y Or objetive is to find the form of the fntion mv? For that prpose, let s onsider a ollision between two idential partiles. Collision between two idential partiles Same ollision seen from a rotated ais Fig.5 The same ollision observed from two different referenes. Let s frther eploit the symmetry of the problem Let s view the ollision from a referene O that is moving horizontally towards the right with respet to the green referene in the figre above at the same speed of the horizontal veloity omponent of partile. The reslting sitation, seen by O, is shown in figre.6 below. 4

15 O v v O v v w w w w Fig.6 Collision viewed in the referene O where the horizontal veloity of partile- is zero. The two graph above are idential. With respet to observer O, we have the following w: vertial veloity of partile-. v: magnitde of the inident and reeding veloity of partile-. : horizontal veloity omponent of partile- Conservation of the horizontal omponent of the linear momentm Notie, de to the symmetry of the problem and sine the partiles are the same before and after the ollision, that the horizontal omponent of the total momentm is onserved. In effet, in the referene O we have see Fig.6: Before the ollision: p = [m v + 0 After the ollision: p = [m v +0 That is, we verify that the horizontal omponent of the total linear momentm is onserved even thogh we do not know yet the epliit form of mv. Conservation of the vertial omponent of the linear momentm What abot the vertial omponent p y? How to apply the onservation of linear momentm? 5

16 First, we need to determine the vertial omponent v y of partile see graph below. We do not know v y. O v v y w w Fig..7 Collision as seen by observer O. The omponent v y needs to be determined in terms of v,, and w. We wold be tempted to say that v y = w, bt that wold be inompatible with the Lorentz transformation, as we will see below. To find the proper relationship between v y and w let s view the ollision in a referene O in whih observer O is seen to travel at speed. De to the symmetry of the problem, the ollision in the referenes O and O will look like in the Fig..8 below where, for omparison, the diagram of figre.7 is repeated. Notie that de to the symmetry of the problem, the diagram observed in O is simply a mirror of the diagram observed in the referene O O O w w O v v y Collision as observed in referene O. Observer O moves with speed relative to O. w w Collision as seen by observer O. 6

17 Fig.8 The same ollision observed in referenes O left diagram and O right diagram. Referene O moves to the right with veloity relative to observer O. The diagrams in Fig.8 allows finding v y in terms of w. y y : Indeed, let s se epression 5 For the motion of partile we identify: Aording to referene O : Aording to referene O : This gives, Ths y = w = - and y y w. y = v y y w 9 Let s retrn now to the diagram displayed in Fig..7, and smmarize or reslts: v y = W O mv v y v w w mw Fig.9 Collision diagram showing the vales of speeds ompatible with the Lorentz transformation. 7

18 Now we apply the onservation of vertial omponent of the linear momentm [mv v y [mw w Partile Partile [ mv w [ mw w, whih gives, mv mw Partile Partile 0 Now the qestion is: What shold be the form of the fntion m sh that it satisfies the relationship 0? Case w~0 To gain some grasp abt the potential soltion for the veloity dependent mass m satisfying 0 let s onsider the ase in whih w~0 glaning ollision. When w0, whih also implies v, we obtain from 0, m0 m Case: Arbitrary sitation Based on the reslt, obtained for the partilar ase of a glaning ollision, what abot onsidering that that epression is indeed the general relationship between mass and veloity? In other words, wold the epression mv m0 v be the soltion that satisfies eqation 0? 8

19 The answer is affirmative and it is left as an eerise see homework- assignment...c. Derivation of the relativisti energy 5 Aording to the fndamental reslt from lassial physis, one obtains, f K F d i f i f i total d mv d dt K W hange in kineti energy eqal work done by the total fore ating on the partile dv dm m v d dt dt 3 mo Bt, from the relativisti mass epression m v, one obtains, Or, m m v mo m Hene, v dm dt dm dt m v m dm mv dt v dm dt o dv m dt dm vm dt dv vm v v dt dm dt and sing d= v dt v dv 0 dt dm dt dv m dt 9

20 dm dv dm v m 4 dt dt d Conseqently, replaing 4 in 3, K f i dv m dt dm v d dt f i dm d d f K dm m f mi 5 i Notie that, for v<<, epression 5 gives K ~ mov see net setion. This makes plasible to interpret K in 5 as the relativisti kineti energy. K v m v mo 6 Etra energy aqired by a partile of initial mass m o pon the ation of an eternal fore Einstein interpreted with veloity v. m v as the total energy of a partile moving E v m v 7..C.d Eqivalene of mass and energy E m How mh does the mass of a partile inreases when its new speed is mh smaller than? m m m o [ v v [... m ~ o o ~ m o [ v... KE Change in the Newtonian kineti energy 0

21 KE That is, m ~. This observation led Einstein to the more general sggestion that E= m. For instane, the epression above an be rewritten as, m = m o + ½ m o v + Interpreted as the total energy of the partile E= m Intrinsi rest energy This theory of eqivalene of mass and energy E= m has been verified by eperiments in whih matter is annihilated; that is mass at rest is onverted totally to radiant energy: An eletron and a positron ome together at rest, eah with a rest mass m o. When they ome together they disintegrate and two gamma rays emerge, eah with the measred energy of m o. Energy has inertia Consider an inelasti ollision [mw w where Before m w mw w m 0 w After M0 Energy= [mw Energy= [M0 Conservation of energy implies

22 M0 = mw > m0 83 Notie that, even thogh the two individal masses ome to rest after the ollision, the mass after the inelasti ollision M0 is greater than [m0: M0 > m0 [mw-m0 is the kineti energy broght in. 94 Bease of the kineti energy involved in the ollision, the reslting objet M0 will be heavier. M0 > m0 When we pt the two masses together gently they make something whose mass is m o ; when we pt them together foreflly, they make something whose mass is greater than m o. This is different than in Newtonian mehanis, where two partiles ollide inelastially and form an objet of mass m o, whih is in no way different form the one reslting ptting them together slowly. There is more kineti energy inside, bt that does not affet the mass. In Einstein s Mehanis, the mass of a system omposed of two partiles depend on how the partiles were broght together gently or violently. However, we an not always identify the parts inside an objet of mass M. M old disintegrate in two, or in three partiles. It is not onvenient, and often no possible, to separate the total energy M of an objet into a rest energy of the inside piees, b kineti energy of the piees, potential energy of the piees; instead we simply speak of the total energy of the partile...c.e Relationships involving p, E and v From the epressions for the relativisti energy in 8 we obtain, mo 4 E mo E or 305 v v From the relativisti momentm epression we obtain,

23 P m o v 36 v E mo Sbtrating 3 from 30 we obtain P v, v whih gives, E P m o 37 m On the other hand, from P m v one an obtain P v, or E v P 338..D For-omponents vetors and symmetry..d.a Transformation of spae-time oordinates Notie the relativisti transformation of oordinates given in t t, y y, z z, t an be epressed also as, t t, y y, z z, t 34 If we denote the oordinates as =, =y, 3 =z, o = t, the transformation of oordinates,, 3, 0,, 3, o adopts a more symmetri form, o ; o ; 3 3 ; o 35 3

24 4..D.b Transformation of the energy-momentm oordinates Starting from the transformation of veloities y y z z one an obtain it is left as an eerise 363 Note: Try to obtain 36 on yor own. Yo may want to hek yor answer with the one given at the end of this hapter. The first energy-momentm transformation is obtained from the epression m m o. Together with 3 it beomes, [ P m m m o Hene P m m E, or P E E 373 Another energy-momentm transformation is obtained sing

25 P m o m. With the help of 36 and one obtains, P P 3833 E The remaining two epressions are more straightforward to demonstrate P y P y P P, z z 3934 They are obtained from, for eample, sing mo Py m y y and then epression 36, et. Note: Yo are enoraged to verify on yor own those last epressions. In smmary, P, P y, P z, m P, P y, P z, m P P m, P y Py P P m P m 40, z z, Notie the similarity with the oordinates transformation, y, z, t t,, y, z, t t t, y y, z z, t The symmetry is more straightforward sing the variables m and t, 5

26 6 P, P y, P z, m P, P y, P z, m m P P, y P y P, z z P P, P m m 4, y, z, t t,, y, z, t t, y y, z z, t t APPENDIX- Optional proedre to obtain epression 36. Starting from the veloities transformation, ; y y ; z z A. Demonstrate [ [. B. Demonstrate [ [, or eqivalently. Soltion

27 7 [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ o From the last epression one obtains, [ [ [ Or, [ REFERENCES This desription follows losely Feynman Letres, ol - I, Chapters 5 and 6. See also Appendi A of this orse s tetbook R. Eisberg and R. Resnik, Qantm Physis, 3 Ref Feynman Letres ol - I, p J. D. Jakson, Classial Eletrodynamis, 3 rd Ed.; John Wiley and Sons; page 56 5 R. Eisberg and R. Resnik, Qantm Physis, nd Edition, Wiley, 985; page A-5 6 Ref. 4, page 533.